Arbitrary high order numerical methods for time-harmonic acoustic scattering problems originally defined on unbounded domains are constructed. This is done by coupling recently developed high order local absorbing boundary conditions (ABCs) with finite difference methods for the Helmholtz equation. These ABCs are based on exact representations of the outgoing waves by means of farfield expansions. The finite difference methods, which are constructed from a deferred-correction (DC) technique, approximate the Helmholtz equation and the ABCs, with the appropriate number of terms, to any desired order. As a result, high order numerical methods with an overall order of convergence equal to the order of the DC schemes are obtained. A detailed construction of these DC finite difference schemes is presented. Additionally, a rigorous proof of the consistency of the DC schemes with the Helmholtz equation and the ABCs in polar coordinates is also given. The results of several numerical experiments corroborate the high order convergence of the novel method.
The formulation of high order local absorbing boundary conditions in terms of farfield expansions (FE-ABC) for time-harmonic waves is considered. Our previously constructed FE-ABC for single and multiple acoustic scattering are improved. Furthermore, we extend the FE-ABC to elastic scattering. By decomposing the vector elastic displacement in terms of scalar potentials, the Navier's equation of elasticity is reduced to a boundary value problem consisting of two Helmholtz equations coupled through their boundary conditions. As a result, the formulation of the elastic FE-ABC from the acoustic case becomes natural. In this work, we present some numerical results by coupling the FE-ABC with finite difference methods and a general curvilinear finite element method based on isogeometric analysis. We present our results for two and three dimensional acoustic and elastic scattering problems from obstacles of arbitrary shape.
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